Scaffolding Close Reading of Mathematical Text in Pre-service Primary Teacher Education at the Tertiary Level: Design and Evaluation

Even in the digital age, learning mathematics at an academic level still requires much reading of mathematical text. Research has shown that reading mathematical text requires readers to engage with all the structures of the book and with its pedagogical voice, making connections, and plausible reasoning. Specific practices and strategies that support the close reading of mathematical text have been suggested; however, descriptions and empirical evaluations of materials designed to support these activities are rare. We present the design and first evaluation cycle of materials developed in a design research project that aims to scaffold close reading of mathematical text. The materials were designed and evaluated in a German university course on elementary geometry for first-year teacher education students who study mathematics to become primary teachers. The reading strategies were explained and modeled for students in reading-strategy videos. Additionally, close reading of mathematical text was scaffolded by close-reading tasks and homework tasks and problems that build on the reading strategies and were specifically designed to foster understanding of the mathematical text. Survey data were collected from 296 students to evaluate their use of and attitude toward the different materials. The quantitative results indicate that students used the materials and were generally able to learn the course content by themselves. From all provided materials, they found the close-reading tasks most helpful. A qualitative analysis of answers to open questions revealed issues with different materials, particularly with the script, and requests for additional materials. The issues with the script were categorized inductively. The categories are presented as a qualitative result of the study and discussed. Supplementary Information The online version contains supplementary material available at 10.1007/s10763-022-10309-y.

Visualization II (based on building 3d-models or projections of 3d-models in GeoGebra) 10. Symmetry of (dual-) Archimedean solids Symmetry of Archimedean solids and their dedicated dual-Archimedean solids.
Visualization III (based on operating with given 3dmodels) * focus in this paper Theorem: The product of two reflections Sg (first mapping) and Sh (second mapping) is: (i) the translation V2•a (with length 2·a) (with the identity as a special case, if g = h), if g and h are parallel and have the (oriented) perpendicular a (vector with length a≥0) from g to h.
(ii) the rotation DZ; 2•w (with the "point reflection" as a special case for w = 90°), if g and h are not parallel, but have a well-defined intersection point Z and enclose the (oriented) angle with measure w ≠ 0 ° (between 90° exclusive and 90° inclusive) from g to h.
(iii) the identity, if g = h (special case of (i) with a as the zero vector, i.e. a = 0; can also be treated like (ii) with g = h, i.e. w = 0 ° and some center Z).
(iv) If you reverse the order of the first and the second mapping, then V-2•a or DZ; -2•w.
! Understanding mathematical statements and argumentations while-reading.

Conclusions:
(i) In general, a composition of two reflections is not commutative. In only two cases Sh • Sg = Sg • Sh: If g=h it is the identity, and if g and h are orthogonal, then the composition is both times a rotation by 180° (=point reflection), and indeed one order equals a rotation by 180°, the other by -180°, which is the same.
The composition of three or more reflections is associative, that means Si • (Sh • Sg) = (Si • Sh) • Sg.
(ii) Let g and h be two parallel straight lines with the (oriented) perpendicular a from g to h. Let k and l be two more parallel lines, also with the (oriented) perpendicular a from k to l (so k and l are parallel to g and h). Then Sl • Sk is equal to the translation V2•a as Sh • Sg, so Sl • Sk = Sh • Sg.
Conversely, if you have a translation Vb , you can substitute it by the product of two reflections Sh • Sg. The two axes need to be chosen as follows: They have to be perpendicular to b, that means in particular parallel to each other; and the length of the perpendicular from g to h must be ! " .
-If these conditions are met, then the axes can be anywhere in the plane; -both reflections, performed in the correct order, result in the given translation.
(iii) Let g and h be two non-parallel straight lines with the intersection Z and measure w of the (oriented) angle from g to h ( -90°<w<90° , w≠0° ). Let k and l be two other nonparallel lines, also with the intersection Z and the same measure w of the (oriented) angle from k to l. Then Sl • Sk is equal to the rotation DZ;2•w as Sh • Sg, Sl • Sk = Sh • Sg.
Conversely, if you have a rotation DY;u , you can substitute it by the product of two reflections Sh • Sg. Besides the case u=0°, the two axes need to be chosen as follows: Both axes have to pass through Y, and the measure of the angle from g to h has to be # " (in particular, it needs to have the same algebraic sign as u). -If these conditions are met, then the axes can lie in any direction through Y; -both reflections, performed in the correct order, result in the given rotation.
(vi) A product Sn • Sn1 • ... • S1 of n reflections S1 , S2 , ..., Sn ( n>3 ) is equal to the product Tm • Tm1 • ... • T1 of m (in generally other) reflections T1 , T2 , ..., Tm with m = n -2. This decrement by 2 reflections can be continued as long as there are more than 3, and finally every product of n reflections is equal to a product of at most three reflections (for even n of at most two) reflections.
The statement in conclusion (vi) is a special case of the three-reflection theorem we already know; it is now proved again constructively for the case of 4 reflections We justify this statement using a configuration of 4 reflections as an example. For the sake of clarity, we do not designate the lines of reflection with lower case letters this time, but number the reflections according to the formulation in the statement above. Therefore, in this case the axes of reflections are designated with the numbers 1-4. S1 denotes the first reflection at axis with the number 1 which is performed in the composition of the four reflections.
Compared to the first image of Fig. I.3.14, S1 and S2 are replaced by S1' and S2' in the second image, preserving intersection point A and angular measure u, in a way that axis 2' passes through intersection point B of axes 3 and 4. In the third image, S3 and S4 are replaced by S3' and S4', preserving intersection point B and angular measure, in a way that axes 3' and 2' are identical.

Abb. I.3.14: Product of 4 reflections is replaced by a product of 2 reflections.
Then S4 • (S3 • (S2 • S1)) = (S4 • S3) • (S2 • S1) = (S4' • S3')•(S2' • S1') = S4' • ((S3' • S2') • S1') = (S4' • I) • S1' = S4' • S1' , -and this is a rotation (fourth image) (in special cases a translation). While the measure of the rotation angle is relatively easy to identify, namely as sum u+v of the angle measures u and v of the two rotations S4 • S3 and S2 • S1, the determination of the center of rotation is more complicated. Here, one has initially the product of two rotations DA;2•u and DB;2•v and obtains as a result the rotation DC;2•(u+v). -For our purposes, it is essential to have succeeded in writing a product of 4 reflections as an equal product of 2 reflections.  Conclusion of the theorem about the product of two reflections in parallel lines "We need two parallel straight lines g and h with the oriented perpendicular a from g to h as well as two further parallel straight lines k and l also with the oriented perpendicular a from k to l." 7:10 -8:25

Rereading
Understand the theorem statement, put a strong emphasis on accuracy Comprehending mathematical statements while reading The product of two reflections in parallel lines f and g is a translation "This statement is completely independent of the position of the two straight lines g and h in the plane.… The